This paper extends and develops the results of a previous paper Malinovskii (2007). Dealing with a simplistic diffusion multi-year model of insurance operations, this paper illustrates the adaptive control approach when the object of control is the balance of solvency and equity. Compared to the previous paper, a new element is the “scenario of nature”, or the incomplete knowledge of future risk, which is quite often the case in insurance. It introduces a new and inevitable randomness in the model and leads to a qualitative difference in its behavior.

We study the continuous-time portfolio optimization problem of an insurer. The wealth of the insurer is given by a classical risk process plus gains from trading in a risky asset, modelled by a geometric Brownian motion. The insurer is not only interested in maximizing the expected utility of wealth but is also concerned about the ruin probability. We thus investigate the problem of optimizing the expected utility for a bounded ruin probability. The corresponding optimal strategy in various special classes of possible investment strategies will be calculated. For means of comparison we also calculate the related mean-variance optimal strategies.

The classical treatment of stochastic models in non-life insurance is to first derive the well-known Poisson distribution by considering the question of how many claims take place during a definite period t. In deriving this distribution the following three assumptions are made:

( I) The population studied is homogeneous.

( II) The occurrence of a claim is a rare event, viz. in an infinite-simal time interval [t, t + Δt], the probability of more than one occurrence must be of the order of magnitude o(Δt).

(III) The occurrence of any later claim is not influenced by previous ones (no contagion).

In my purely theoretical study [5]), the consequences of discarding one or more of the above assumptions were considered. By so generalizing the Poisson distribution, a great many stochastic models can be built, although the results were not always successful. The following study concentrates on some considerations based on assumption (II). The theoretical formulation of the model will be dealt with briefly and the author would first make reference to the instructive article of Thyrion [7] which was unfortunately unknown to him when he was preparing his already mentioned paper. Ammeter [2] and Arfwedson [3] have also considered special cases of this generalization. With the help of the statistics over traffic accidents in the city of Zurich, I hope to throw some more light on the practical aspects of the problem. To the Statistics Office of this city I would express my thanks for kindly placing all documents at my disposal.

The multiplicative ratemaking, model we have in mind is the following one. Within a certain branch of insurance we have, say for simplicity, two tarif arguments U and V. For example, in motor insurance we could think of U and V as being make of car and geographical district respectively. In fire insurance U could be class of construction for buildings and V could relate to fire defense capacities.

The arguments are of a qualitative nature and argument U has r levels, while argument V has k levels. To our disposal we have statistical experience of the business for a certain period of time, consisting of

—risk exposures nij (i = 1 … r, j = 1 … k).

Risk exposure nij thus corresponds to the ith U-level and the jth V-level. It could be e.g. number of policy years or sum insured during the period of observation for objects belonging simultaneously to U-level i and V-level j.

The nijS are known non-random quantities.

—(relative) risk measures pij(i = 1 … r, j = 1 …k).

Risk measure pij could be e.g. claims frequency, i.e. number of Claims divided by number of policy years, or claims cost per policy year or claims cost as a percentage of sum insured. In general pij is thus the observed number or the observed amount of claims belonging simultaneously to U-level i and V-level j, divided by the corresponding risk exposure nij.

One of the most fundamental tasks in non-life insurance, done on regular basis, is risk reserving assessment analysis, which amounts to predict stochastically the overall loss reserves to cover possible claims. The most common reserving methods are based on different parametric approaches using aggregated data structured in the run-off triangles. In this paper, we propose a rather non-parametric approach, which handles the underlying loss development triangles as functional profiles and predicts the claim reserve distribution through permutation bootstrap. Three competitive functional-based reserving techniques, each with slightly different scope, are presented; their theoretical and practical advantages – in particular, effortless implementation, robustness against outliers, and wide-range applicability – are discussed. Theoretical justifications of the methods are derived as well. An evaluation of the empirical performance of the designed methods and a full-scale comparison with standard (parametric) reserving techniques are carried on several hundreds of real run-off triangles against the known real loss outcomes. An important objective of the paper is also to promote the idea of natural usefulness of the functional reserving methods among the reserving practitioners.

Stimulated by the paper of W. S. Jewell) on multidimensional credibility I should like to show that at level 2 assumptions (Jewell's terminology) one can obtain explicit formulae for forecasting total losses in the future based upon total losses and number of claims observed in the past.

En Suisse, les sociétés d'assurances bonifient à leurs assurés en responsabilité civile autos une ristourne qui dépend du nombre des années sans sinistres, suivant le barème ci-après:

Cette ristourne a pour but de tenir compte des facteurs d'ordre subjectif qui influencent le risque et qui ne peuvent être pris en considération dans les tarifs.

Les deux questions suivantes se posent:

A) Y a-t-il en réalité des différences dans le risque subjectif justifiant la bonification d'une ristourne pour années sans sinistres?

B) Si oui, le barème indiqué ci-haut tient-il compte de ces différences d'une façon adéquate?

Le présent travail a pour but de répondre à ces deux questions en appliquant les méthodes modernes des tirages au sort.

Question A

Voici les résultats de quelques tirages au sort du portefeuille de „La Générale de Berne”

S'il n'y avait pas de différences dans le risque subjectif, la variable aléatoire k devrait suivre la distribution de Poisson. Toutefois, on sait que dans ce cas, devrait être égal à s2(k), ce dont il n'est évidemment pas question pour les chiffres ci-dessus. En conséquence, k n'est pas distribué d'après Poisson, et l'existence de différences dans le risque subjectif est prouvée.

On pourrait se demander si les différences entre et s2(k) sont de nature aléatoire. Une investigation spéciale a toutefois montré que si k suivait la distribution de Poisson, la probabilité pour des différences aussi élevées que celles des tirages indiqués ci-haut serait pratiquement nulle.

We derive some decision rules to select best predictive regression models in a credibility context, that is, in a “random effects’ linear regression model with replicates. In contrast to usual model selection techniques on a collective level, our proposal allows to detect individual structures, even if they disappear in the collective.

We give exact, non-asymptotic results for the expected squared error loss for a predictor based on credibility estimation in different models. This involves correct accounting of random model parameters and the study of expected loss for shrinkage estimation. We support the theoretical properties of the new model selectors by a small simulation experiment.

We analyze and compare the impact of tax incentives and of introducing enhanced annuities on annuitization behavior considering heterogeneity among the insured. We find that tax incentives for annuitization result in a significant increase of the portion of people who should annuitize and also an increase of the insurer's profit since less healthy individuals also annuitize, i.e. adverse selection is reduced. However, the problem that different insured receive a different value for money is even increased by tax incentives. If enhanced annuities are introduced, the percentage of insured who should annuitize further increases. Adverse selection is further reduced and the differences in value for money from annuitizing shrink.

We propose a new parametric model for the modelling and estimation of event distributions for individuals in different firms. The analysis uses panel data and takes into account individual and firm effects in a non-linear model. Non-observable factors are treated as random effects. In our application, the distribution of accidents is affected by observable and non-observable factors from vehicles, drivers and fleets of vehicles. Observable and unobservable factors are significant to explain road accidents, which mean that insurance pricing should take into account all these factors. A fixed effects model is also estimated to test the consistency of the random effects model.

Dans un traité antérieur, l'auteur a examiné, en collaboration avec M. Segerdahl, un certain nombre de fonctions analytiques propres à définir la distribution du montant d'un sinistre en s'arrêtant tout spécialement aux aspects présentant de l'intérêt du point de vue de l'assurance en excédent de sinistres. L'examen a révélé que les répartitions du type Pareto étaient intéressantes aussi bien du point de vue théorique — en tant que représentant un type de distribution particulièrement „dangereux” — que du point de vue pratique, le matériel statistique suédois et norvégien obtenu lors de l'analyse de différentes branches d'assurances — telles que Vie, Incendie et Automobile — semblant bien indiquer que pour de grands intervalles la distribution du type Pareto représenterait l'authentique matériel des dommages.

Nous nous permettrons par conséquent de nous arrêter tant soit peu aux caractéristiques de la distribution du type Pareto ainsi qu'aux formules découlant de la dite distribution pour la prime de risque en excédent de sinistres et ses variances.

Si P(x) exprime la distribution, I − P(x) = H(x) = C. x−∝ pour a < x ≤ M, M étant la valeur la plus grande pour x, en supposant que la probabilité d'atteindre la valeur M est H(M) = C. M−∝.

Si m(x) est la moyenne des portions de sinistres supérieures à x

Spécialement si M = ∞ on obtient .

In case of a stop-loss treaty the reinsurer takes over that part of the risk that exceeds a given amount y1. We will deduce bounds on a modified stop-loss treaty where the liability of the reinsurer is limited to y2–y1 in case the claim amount exceeds y2. Upper and lower bounds of this modified stop-loss premium are obtained as a simple application of results obtained earlier by the first author.

In a note on the security loading of excess loss rates I am deducing a simple formula intended to replace some tedious calculations. In the beginning of that note I made the point that some authors recommend a loading proportional to the dispersion of the total claims amount of a treaty δ1 while others tend to favour .

I also stated that a loading proportional to δ1 or its estimate δ1* could be deduced from the statistical uncertainty in measuring the risk (section 4).

The question has been raised if and to what extent a loading system based on the dispersion is unduly punishing the smaller portfolios. This will be examined below.

The pricing concept will be analyzed from the point of view of a big dominating Reinsurer who wants to be fair in all directions. The conclusion of this study supports an affirmative answer to the question put above.

In a second part the loading is studied from a different angle bringing competition into the picture. The pricing or loading becomes a problem of operations research under the simplified assumption that profit is the only purpose of our activity. Not unexpectedly, the loading coming out from this aspect differs from those of part one.

Part two also deals with the question of how much of the loadings which we are aiming for, get lost in the competitive process. It is also shown that in most cases the harder the competition is, the higher loadings shall be used.

Part one and part two thus deal with the loading problem from different aspects, and illustrate the complexity of the problem. It is my hope that this note could stimulate further researches in this interesting and important area, also in a moment when some reinsurers are more concerned with the question of surviving than in fixing the loadings which should on the average and in the long run turn up as profits.

This paper investigates the effects of high or low fair-premium demand elasticity in an insurance market where risk classification is restricted. The effects are represented by the equilibrium premium, and the risk-weighted insurance demand or “loss coverage”. High fair-premium demand elasticity leads to a collapse in loss coverage, with an equilibrium premium close to the risk of the higher-risk population. Low fair-premium demand elasticity leads to an equilibrium premium close to the risk of the lower-risk population, and high loss coverage – possibly higher than under more complete risk classification. The demand elasticity parameters which are required to generate a collapse in coverage in the model in this paper appear higher than the values for demand elasticity which have been estimated in several empirical studies of various insurance markets. This offers a possible explanation of why some insurance markets appear to operate reasonably well under community rating, without the collapse in coverage which insurance folklore suggests.

Systemic risk (SR) is considered as the risk of collapse of an entire system, which has played a significant role in explaining the recent financial turmoils from the insurance and financial industries. We consider the asymptotic behavior of the SR for portfolio losses in the model allowing for heavy-tailed primary losses, which are equipped with a wide type of dependence structure. This risk model provides an ideal framework for addressing both heavy-tailedness and dependence. As some extensions, several simulation experiments are conducted, where an insurance application of the asymptotic characterization to the determination and approximation of related SR capital has been proposed, based on the SR measure.

Pricing approaches for long-tail quota shares are often based on the chain ladder method. Apart from IBNR calculation, common pricing methods require volume measures for accident years in the observation period, and for the quotation period. In practice, in most cases restated premiums are used as the volume measures. The prediction error of the chain ladder method is an important part of the prediction uncertainty of these pricing approaches. There are, however, two sources of uncertainty that are not addressed by the chain ladder model: the stochastic volatility of the claims in the first development year; and the restatement uncertainty, the risk that the restated premium is not a good volume measure. We extend Mack's chain ladder model to cover these two sources of uncertainty, and calculate the mean-squared error of chain ladder pricing approaches with arbitrary weights for the accident years in the observation period. Then we focus on the problem of finding optimal weights for the accident years. First, we assume that the parameters for restatement uncertainty are given, and provide recursion formulas to calculate approximately-optimal weights. Second, we describe a maximum likelihood approach that can be used to estimate the restatement uncertainty.

Stop Loss reinsurance has attracted the interest of ASTIN members for years. May I recall the paper of Borch [1] in which he demonstrates some optimality qualities of the stop loss reinsurance from the ceding company's point of view, the contribution of Kahn [2] and the paper of Pesonen [3]. I also mention the paper of Esscher [4] and Verbeek's contribution [5]. Going back to the pre-ASTIN days we find a paper of Dubois [6].

The rating problems have been dealt with by several authors. Let me recall the rating formula worked out by a group of Dutch Actuaries some 20 years ago. This was based on the assumption that the mean and the standard deviation were known. Based on Chebycheff's inequality an approximation formula was worked out which, of course, was heavily on the safe side.

Even younger members of ASTIN are probably familiar with the studies made in the early sixties by a group of Swedish Actuaries, the results of which were presented by Bohman at the Actuarial Congress in London in 1964. Partly based on this, Bühlmann worked out some tables which he used for rating purposes.

My present contribution to the subject may not justify the above reviews, particularly as I will deal with a very special retention situation which a practical underwriter will rightly not accept, namely a stop-loss point as low as equal to the mean value of the distribution.

My excuse for this is that the formula deduced is very handy and that it is of value to the underwriter to know the stop loss risk rate also at this low level.

In the past the subject of optimal reinsurance has been dealt with by various authors such as Borch (1), Kahn (2) and Verbeek (3). Borch tries to look at the problem from the point of view of a company which acts as both Insurer and Reinsurer whereas the viewpoint of Kahn and Verbeek is that of a Ceding Company.

Let us study the simplest possible market situation of one Insurance Company C and one Reinsurer R. C writes a portfolio P and seeks reinsurance protection at R.

In the present note I will start with the viewpoint of optimizing simultaneously the situation of C and R.

For both C and R the variance is used as a measure of risk. The variance of P is V. C makes an effort to reduce V to a level Vc by way of a reinsurance treaty T. The corresponding variance of R is VR.

Out of the set of possible reinsurance treaty arrangements {T} we assume that the parties are first of all interested in such arrangements for which

In other words they will look for solutions which will lead to a substantial reduction of the variance.

It is possible to think of “nonsense” arrangements where no substantial reduction is achieved and in the extreme case even of arrangements for which

In a further paper we might deal with such a case.

We study a weighted comonotonic risk-sharing problem among multiple agents with distortion risk measures under heterogeneous beliefs. The explicit forms of optimal allocations are obtained, which are Pareto-optimal. A necessary and sufficient condition is given to ensure the uniqueness of the optimal allocation, and sufficient conditions are given to obtain an optimal allocation of the form of excess of loss or full insurance. The optimal allocation may satisfy individual rationality depending on the choice of the weight. When the distortion risk measure is value at risk or tail value at risk, an optimal allocation is generally of the excess-of-loss form. The numerical examples suggest that a risk is more likely to be shared among agents with heterogeneous beliefs, and the introduction of the weight enables us to prioritize some agents as part of a group sharing a risk.

This article presents an explicit formula for the value of a withdrawal benefit when the times of death and withdrawal are dependent. The derivation is based on an actuarial equivalence principle. As a special case, we show that in the fully continuous case, the withdrawal benefit is the reserve when the decrements are independent. We also present a definition of antiselection and prove that the withdrawal benefit will be smaller under antiselection.